Table Of ContentA Gateway to
MODERN
MATHEMATICS
Adventures in Iteration II
Universities Press
Ww
SHAILESH A SHIRALI
the RMS series Little
The books inthis series
school students who are
mathematics a ite
ol curriculum offers.
Iterations isan exciting topic of study and should
interest. both the amateur as well as. the
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Shailesh A Shirl served a
Principal of Rishi Valley School
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003, He has a deep interest in
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LITTLE MATHEMATICAL TREASURES
VOLUME 2
A Gateway to
MODERN
MATHEMATICS
Adventures in Iteration II
SHAILESH A SHIRALI
Rishi Valley School
Andhra Pradesh
w
Universities Press
Contents
Preface xi
Review ait
1. Insights from calculus 1
1.1 Error estimates 2
1.2 Attraction and repulsion 3
1.3 What if the slope is 1 or 1? 4
1.4 Applications 5
1.5 Comments on cobwebbing 2
1.6 Convergence to 2-cycles 1B
1.7 Contraction mappings, 3B
18 An iteration for x 16
1.9 Exercises 18
2. Solution of equations 19
2.1 ‘The Newton—Raphson algorithm 20
2.2 Examples 2
2.3 Comments and extensions 4
24 Halley's method 6
2.5 The method of false position 29
2.6 A route to square roots 31
2.7 A route to cube roots 34
2.8 Exercises 36
3. Atower of exponents 37
3.1. An infinite exponential 37
3.2. Number crunching I 37
3.3 The number e 40
4
3.4 Explanations
“a
35 Number crunching It
3.6 Maximising x'/*
3.7 The occurrence of 2-cycles
38 The outcome when's = el/@
3.9 The outcome when 0.<u<e~*
3.10 The outcome when u = e~*
3.11. Number crunching IL
3.12 Exercises
‘The drifters
4.1. The iteration x (22 —1)/2e
42 Explanations
43 The iteration x —+ ex(1-2)
44 Explanations
45 Period doubling forthe iteration x—+ ex(1 ~2)
46 Exercises
. Memorable problems from the Olympiads
5.1 Problem IMO 1986/3
5.2. Problem IMO 1993/6
53 Ex
54 Two USAMO problems
‘555 An almost-ran IMO problem
56 Exercises
6. Miscellaneous problems
6.1 Four problems
62 Solutions
63 Ramanujan’s problem
{64 A problem from the CMJ
65 Another GCD iteration
66 The Mersenne iteration
6.7 Exercises
Contos &
7. Sarkovskii’s theorem, us
7.1 Proof of the theorem of Li and Yorke 18
8, Estimating the speed 13
81 Theiteration x + x— 2 ps
82 Theiteraions—+2—2 17
83 The iterated sine 128
84 Theiteraion s+ 2-4 1/x 132
85 The Tower of Exponents 133
86 Exercises 136
9. Fermat's two-squares theorem 137
9.1 Zagier’s proof 137
9.2 A constructive proof 141
9.3 Proof ofthe algorithm 12
94 Exercises 144
10. Graphics through iteration 146
10.1 The Cantor set 146
102 The Sierpinski gasket 149
103 The Chaos Game 150
1044 erated function systems (IFS) 152
105 Turtle walk 155
106 Exercises 163
11, Fractional linear maps over C 164
11.1 Functions defined on the extended complex plane 164
11.2 Linear mappings 166
112.1 The case a= 1 166
11.22 The case a1 166
11.23 The case when ais a root of unity 169
11.3 Mobius transformations 170
114 Iterations under Mobius transformations 13
IS Exercises 182
R
13.
‘Quadratic maps over C
12.1 Sample orbits of the map z+ 2? +e
122 Exercises
‘The Mandelbrot set
131 Julia—Fatou sets
132 Iulia set for the logistic map
133 The Mandelbrot set
134 Shape ofthe Mandelbrot set
135 Exercises
136 MATHEMATICA programs
Appendix A
Appendix B
Appendix.
Appendix D
Index
Preface
‘This book isa sequel to the author's A Gateway to Modern Mathematics: Adventures
Iteration, Volume I (which we shall refer to as Adventures 1). The idea of iteration
‘was introduced in that work, together with various associated notions (fixed points,
‘orbits, cycles, limit points, convergence, solution of equations, cobwebbing, and so
‘on), and a large number of examples were studied from the world of arithmetic,
algebra and geometry. The present work continues the study of iteration, but at a
higher level. However, itis largely self-contained, and can be read without reference
to Adventures I
‘Students who are preparing for the Regional Mathematics Olympiad (RMO) and
the Indian National Mathematics Olympiad (INMO) will find this book useful. It
is suitable for self-study by students in the age range 15-21 years (this includes
students of mathematics atthe college evel, who will enjoy seeing “in action"” some
of the facts learnt in their early courses in calculus and analysis), and for lay readers,
‘who will enjoy learning about a topic which is of great current interest. It can also
‘be used by teachers in a school mathematics club.
Preview of book
[At the start of the book, in the unnumbered ‘‘Review"” chapter, we present a very
brief review of the material covered in Adventures I
Following this, in Chapter 1, we examine the insights on iteration provided
by differential calculus. Then, in Chapter 2, we study various approaches 10
the numerical solution of equations (c.g., the Newton—Raphson method). These
approaches, as also the iterations studied in Chapters 3 and 4, offer very good
‘examples for illustrating the applications of calculus. In Chapters 5 and 6, we tackle
‘an assorted list of problems. Some of these have their origins in the mathematical
‘olympiads, but we also discuss in detail a well-known problem of Ramanujan's that
‘was once used in the Putnam Lowell competition, and an open problem related 10
“Mersenne primes. In Chapter 7, we give brief account of two fascinating discoveries
‘made in recent years—the theorems of Li and Yorke, and of Sarkovskii. In Chapter 8,
‘we study the speeds of convergence in iterative processes. Here some knowledge
is required of the use of L'Hospital’s rule for computing limits. In Chapter 9, we
consider the well-known two-squares theorem of Fermat concerning primes of the
form 4n-+ 1; we prove it using an iterative technique, and in the process arrive at a
constructive proof, In Chapter 10, we study how recursion makes for a simple and
clegant construction of certain tree-like structures which have an amazingly “real”
appearance. Finally, in Chapter 11, 12 and 13, we step into the complex plane, and
discuss Julia sets of fractional linear maps and quadratic maps, and some associated
notions, including the famous Mandelbrot set.
Price
‘There are plenty of problems scattered throughout the book, with answers or
solution to all except a few given atthe end of the book.
Appendices
‘Appendices A and B have brief reviews of analysis and calculus (notions of limit,
continuity, slope, derivative, etc.) and the algebra of complex numbers (notions
‘of magnitude and argument, de Moivre's theorem, etc), respectively; Appendix C
ives a list of books and internet sites for further reading: and Appendix D gives
solutions to many of the problems in the book.
Prerequisites
‘The book presupposes some degree of mathematical maturity on the part of the
reader—an elusive quality that means, roughly, the ability to rhink mathematically.
It's the earmest and committed belief ofthis author that working through this book
will itself greatly enhance te learner's mathematical maturity.
Acknowledgements
Grateful acknowledgements of thanks are due, for the assistance that I received
from numerous individuals: to Professor Phoolan Prasad, Dr. Jayant Kirtane and
‘Mr. Siddhartha Menon, for going through early versions of the book and offering
helpful and critical comments; to Mr. and Mrs, Mohan Subramaniam, for their
help, generously given, in procuring the IIEX software; to Mr. K Seshadri, for
technical help withthe computer systems; to Professor E Sampathkumar, for his
enthusiastic supportin publishing this book; tomy friends inthe “Olympiad cel,” C
I Pranesachar ("CRP"), J Venkatachala “BJV") and C'S Yogananda ""CSY"
‘with whom I have had numerous iterated adventures; and to Padmapriya, my wife,
for her iterated support.
